Avalanche correlations in the martensitic transition of a Cu-Zn-Al shape memory alloy: analysis of acoustic emission and calorimetry

Avalanche correlations in the martensitic transition of a Cu-Zn-Al shape memory alloy: analysis of acoustic emission and calorimetry Jordi Bar´o1, Jos´e-Mar´ıa Mart´ın-Olalla2, Francisco Javier Romero2, Mar´ıa Carmen Gallardo2, Ekhard K.H.Salje3, Eduard Vives1and Antoni Planes1. 1Departament d’Estructura i Constituents de la Mat`eria. Facultat de F´ısica. Universitat de Barcelona. Diagonal, 647, E-08028 Barcelona, Catalonia, Spain 2Departamento de F´ısica de la Materia Condensada, Universidad de Sevilla, P.O. Box 1065, E-41080 Sevilla, Spain. Department of Earth Sciences, University of Cambridge, Downing Street, Cambridge CB2 3EQ, United Kingdom. Abstract. The existence of temporal correlations during the intermittent dynamics of a thermally driven structural phase transition is studied in a Cu-Zn-Al alloy. The sequence of avalanches is observed by means of two techniques: acoustic emission and high sensitivity calorimetry. Both methods reveal the existence of event clustering in a way that is equivalent to the Omori correlations between aftershocks in earth quakes as is commonly used in seismology. PACS numbers: 81.30.Kf, 64.60.av, 89.75.Da, 91.30.Dk Submitted to: Journal of Physics: Condensed Matter This is the Author's accepted manuscript (AM) version. The Version of Record is available at Journal of Physics Condensed Matter 26 125401 (2014) under doi: 10.1088/0953-8984/26/12/125401 copyright: IOP Publishing Ltd. Avalanche correlations in the martensitic transition 2 1. Introduction In recent years there has been an increasing interest in the study of systems that respond intermittently in the form of crackling noise when driven by an external field [1]. Examples include a variety of physical phenomena among which magnetization processes [2], plastic deformation in solids [3, 4], materials failure [5, 6] or Earth seismicity are worth mentioning [1]. Crackling noise consists of a discrete sequence of events which usually occur in the form of avalanches. Often these avalanches show absence of characteristic scales which reveals the existence of a certain kind of criticality which is usually displayed by the power law distribution of the energies or amplitudes of the avalanches. Except for earthquakes (that have strong implications for our wellbeing), little attention is paid to the study of the time scales over which avalanches occur. Probably this is due to the fact that the existence of time correlations between avalanches is not a necessary condition for avalanche criticality to occur. In fact, it is known that a temporal sequence of independent events may have energies power law distributed as has been shown for the zero-temperature Random Field Ising model with metastable dynamics [7, 8]. In any case, the existence of correlations provides a higher degree of complexity to the process and its analysis is expected to supply relevant information for a better understanding crackling noise phenomena. This is indeed the case of Earth seismicity where the study of waiting times and time correlations between earthquakes has been a subject of interest from the beginning of this research. Martensitic transitions are another example of process that have been suggested to display avalanche criticality [9]. In these transitions avalanches are usually detected from the measurement of the acoustic emission (AE) which originates from displacement discontinuities across propagating interfaces [10]. This sound, typically recorded in the ultrasonic frequency range, carries the whole temporal and spacial information which characterizes the evolution of the internal strain field during the transformation process. Therefore, upon cooling (forward transition) or heating (reverse transition), martensitic systems evolve by relaxing from one metastable state to another within an energy landscape that characterizes the complex multiphase phase coexisting region [11]. It is worth pointing out that not only the statistical distribution of the energy of avalanches can be obtained from AE measurements but also the waiting times between avalanches. Beside AE, other more macroscopic measurement techniques such as high sensitivity calorimetry, [12, 13] have been used in order to show that martensitic transitions do not proceed smoothly but instead occur discontinuously through an avalanche process. The interest of calorimetry is that despite its limited time response compared with the AE technique, it enables to characterize avalanches at a more macroscopic scale at which the heat conduction problem associated with the dissipation/absortion of latent heat during the relaxation between metastable states can be taken into account. That is, with this technique an absolute measurement of the energy released by the discontinuous changes can be undertaken as well as a quantification of the ratio of this released energy over the total latent heat. Acoustic emission, on the contrary, only allows for a relative Avalanche correlations in the martensitic transition 3 measurement of energies of the observed discontinuities, but it is able to resolve many more avalanches on a much finer time scales. The comparison of the data obtained by both calorimetric and AE techniques [13], reinforces the suggestion of a critical distribution of avalanches in martensitic transitions: the energies of the avalanche events {Ek}, averaged over space and time, are distributed according to a power law p(E)∼E−ϵwith ϵ≃2.15 which implies that these systems evolve without characteristic energy scales. The phenomenon thus has many similarities with other complex dynamic critical systems. For instance, by comparing the behavior of martensites with results from seismology, such a power-law distribution of energies can be understood as the equivalent of the Gutenberg-Richter law [15] for the magnitude distribution of earthquakes that can be obtained when seismological data is accumulated from different time periods and different regions on the Earth crust. The comparison can also be extended to the studies of other first-order phase transitions: the distribution of amplitudes of the Barkhausen pulses in ferromagnets driven by an external field show also a lack of characteristic scales and different universality classes have been identified [2]. In this work we focus on the analysis of the correlations between the times of occurrence {tk}of the avalanches during a martensitic transformation. Many tools have been developed within the seismology community in order to analyse correlations between earthquakes. Here, inspired by these techniques, we will show that both measurement techniques, AE and calorimetry, reveal a non-Poissonian character of the transformation process. This implies that there exist small temporal correlations between avalanches. Keeping the comparison with seismology, we will propose a law similar to the so called Omori-law for aftershocks after a main earthquake [16]. A preliminary work [17], based on calorimetric measurements, already suggested the existence of corrections to Poisson behavior in the distribution of waiting times between avalanches, although the data were not conclusive. Here, by using simultaneously AE and calorimetric data we will argue that the correlations do exist. We shall also mention that the analysis presented here differs from and complements those based on Fourier Transformation that have been performed, for instance, for the case of Barkhausen noise in ferromagnets [2]. In section 2 we will present the details of the sample and experimental setup. In section 3 we will present results related to the avalanche rate and avalanche energies. In section 4, the existence of correlations will be evidenced by three different numerical analysis techniques: (A) a direct evaluation of the aftershock activity rate after big events (with energy above a selected threshold), (B) the use of the Bi-test [18] showing the failure of the local Poisson hypothesis and (C) the scaling analysis [19] of the distribution of waiting times δk=tk+1 −tkbetween consecutive events. Finally, in section 5 we will summarize our results and conclude. Avalanche correlations in the martensitic transition 4 2. Experimental The same sample has been used for the calorimetric and AE experiments. It is a polycrystal with composition Cu67.64Zn16.71Al15.65 and a typical grain-size diameter of 1.36 mm. It has a parallelepipedic shape with a height of 3.89 mm and an octagonal base of 0.94 cm2. The mass of the sample is 2.6503 g. More details about sample characterization, preparation and heat treatments can be found in Ref. [13]. It displays a martensitic transition from a cubic Fm3m (L21) high temperature phase to a monoclinic I2/m (18R) structure at low temperatures. The calorimeter [20] consists of two fluxmeters —which are electrically placed in series while thermally in parallel— that press the sample. The fluxmeters are also thermally coupled to a large calorimetric block. The system is cooled down to ensure that the sample is in the low symmetry phase and then heated at a rate Rcal = 0.011 mK s−1(approximately one kelvin per day). During the heating process, the electromotive force provided by the fluxmeters —which is proportional to the heat exchanged by the sample— was recorded by a nanovoltmeter Keithley K2182 at a sampling rate of 12.5 Hz. Hereafter we will refer to these values as calorimetric data. The temperature of the calorimetric block was recorded by using a platinum resistance thermometer. Linearly interpolated temperature values were later assigned to every calorimetric data. High frequency noise in the calorimetric data was filtered out, after the experiment, by a fifth-order all-pole Butterworth filter with a normalized cutoff frequency equal to 8 ×10−2. If calorimetric data are plotted versus time or temperature, a series of bursts can be observed during the martensitic transformation. A typical example is shown in Fig. 1. Note that the burst occur above a continuous background that changes very smoothly with temperature. Both the burst and the background change sign when reversing the thermal driving. It has been suggested that the smooth and the burst response, also classified as jerks, can be associated to two different transformation mechanism [17]. Here we focus only on the analysis of the discontinuous burst that can be treated as avalanche phenomena. We restrict the analysis to a heating ramp. By finding the local maxima and minima, we can evaluate all the positive bursts ∆Vkoccurring at times tk(or temperatures Tk). Quite generally, the time distances between consecutive positive bursts is much longer than their rising times. Thus we identify the bursts as signatures of avalanches and the slow decays as consequences of the response of the calorimeter and the smooth background curve. Assuming an exponential behavior of the instrumental transfer function of the calorimeter, the sharp bursts ∆Vkmeasured in voltage can be translated to avalanche energies Ethrough the relation Ek=∆Vkτ/G where τ= 100 s is the time constant of the calorimeter and G= 130 mVW−1is the gain. The sequence of avalanche energies Ekat temperatures Tk corresponding to the signal in the upper panel are shown in Fig. 1(b). This corresponds to a small set of the full record of calorimetric avalanches studied here that consists of 1655 signals. Avalanche correlations in the martensitic transition 5 The AE setup consists of a PCI-2 acquisition system from Mistras Group (France), working at 2 MHz. The voltage signal from a piezoelectric transducer (R15LT) attached to the upper surface of the sample is amplified (60 dB), and filtered (100 kHz −2 MHz). Individual AE events are defined when the voltage crosses a threshold at 24 dB at time tk. The end of the event (tk+∆k) is determined when the voltage crosses the threshold in the downward direction and remains below threshold for more than 100 µs. The integral of the square voltage during the duration ∆k, normalized by a reference resistance provides a direct estimation of the energy Ekof the signal detected by the transducer. Signals have been recorded during a heating ramp between 220 K to 270 K at a rate RAE = 1.67 mK s−1. Note that the calorimetric ramp has been chosen to be much slower that the acoustic emission one in order to partially compensate for the slow response of the calorimeter. Fig. 1(c) shows an example of the energies Ekat times tkof the avalanches recorded by the AE setup. The full run analyzed in this work consists of 85,460 signals. Note also that the total number of AE signals is much larger than the number of signals extracted from calorimetric data. Moreover the energies of the signals are much bigger for the calorimetric data than for the AE data. It is difficult to compare both energy magnitudes since it is not clear what fraction of the energy of an avalanche is released as acoustic waves and what fraction is released into heat. Moreover, only a tiny fraction of the radiated acoustic energy is recorded by the transducer. We speculate that most probably, only extremely big events or overlaps of many events are recorded by the calorimeter. In the next section we will show that very similar statistical results have nevertheless been obtained by both techniques. This is a robust indication of the existence of criticality: results do not depend on the observation scale. 3. Avalanche rate and energy distribution The avalanche process is not homogeneous during the phase transition. Fig. 2 shows a comparison of the avalanche activity λcal and λAE (expressed as the number of recorded signals per kelvin) as a function of temperature, obtained by calorimetry and AE respectively. Note that the scales for both experimental techniques (left and right) differ by a factor one hundred. Note also that the variations of the rate during the transition show a profile that can be fitted by the superposition of two Gaussian functions and that the experimentally measured values of λcal and λAE span several orders of magnitude. The two curves show a similar behaviour, suggesting a reasonable proportionality. The fact that the second peak (at high temperatures), detected by AE is much smaller than the second peak detected by calorimetry, can be atributed to the high attenuation of the AE signals when the sample is partially transformed. From the analysis of the full sequences we have also computed the distribution of energies p(E) (Gutenberg-Richter-like laws). Fig. 3 shows the normalized histograms corresponding to calorimetry and AE in log-log scale. The straight line corresponds to the exponent ϵ= 2.15 from Ref. [13]. This value is consistent with the present data, Avalanche correlations in the martensitic transition 6 0 0.2 0.4 243.5 243.6 243.7 243.8 243.9 244 E (fJ) T (K) (c) 0 2 4 E (mJ) (b) 8 10 12 14 16 V (mV) (a) Figure 1. (a) Example of voltage signal from the calorimeter during a heating ramp (b) Corresponding sequence of calorimetric avalanches. (c) Acoustic emission avalanches. The correlation with the above sequence cannot be expected since the AE and calorimetric experiments are not performed simultaneously. although other calorimetric experiments have found and slightly smaller exponent [17]. This exponent is not strongly universal since it is known to depend on the symmetry of the low temperature martensitic phase, the driving rate [21] and the driving mechanism Avalanche correlations in the martensitic transition 7 0 10 20 30 230 240 250 260 270 0 100 200 300 lAE (103 K-1) lcal (K-1) T (K) AE cal Figure 2. Activity (number of avalanches above threshold per kelvin) during the heating ramp corresponding to calorimetric avalanches (right scale) and to AE avalanches (left scale). The bins are 0.423 K for AE and 0.436 K for calorimetric data. The dashed lines show fits obtained by the sum of two Gaussian functions. [10]. Note that there are ten orders of magnitude separating the largest AE signals from the smaller calorimetric signals. The apparent vertical offset of the two histograms is due to the fact that both are normalized independently. Both sets of energy values are hardly comparable because one corresponds to the energy recorded from the ultrasonic waves reaching the AE transducer and the other corresponds to the latent heat dissipated by the avalanches. 4. Time correlations Beyond the non-homogeneity of the avalanche activity, which is a characteristic feature of the kinetics of martensitic phase transitions, it has been suggested that after the nucleation of a domain in the cubic phase and during its growth, local heterogeneities in the stress tensor can cause some kind of interdependence between closely recorded signals [22, 23]. A similar phenomenon, reported more than one century ago by Omori [16], is observed in seismology: the occurrence of big earthquakes, called mainshocks (MS), at time tMS can trigger a sequence of aftershocks (AS) in a neighboring region with an activity rate rAS that decays in time as a power-law rAS(t−tMS) = K (c−t−tMS)p,(1) where pis an exponent close to 1 for the case of earthquakes and cand Kare independent of time. Moreover, according to the so called Productivity Law [24] of aftershocks, the stronger the mainshock the more aftershocks it will trigger. Thus the numerator Kis expected to depend on the the energy of the MS as K∝E2α/3 MS with the productivity exponent α≃0.8 (see Ref. [24]). This behavior has been reported in other intermittent processes [25] as well as in numerical simulations [26]. Avalanche correlations in the martensitic transition 8 10-4 10-2 100 102 104 106 108 10-2 10010210410-20 10-18 10-16 10-14 10-12 10-10 10-8 1012 1014 1016 1018 P(E) (aJ-1) E (aJ) e = 2.15 AE cal Figure 3. Energy distribution of AE and calorimetric avalanches during the full heating process. Bins are logarithmic and increase by a factor 3/2. Axes for AE (calorimetry) are left and bottom (right and top). 4.1. Aftershock sequences The analysis of aftershock-like sequences requires a precise definition of an AS within the event series. For our calorimetric and AE data series, we adopt the same definition used within the seismology context, taking into account that we do not have access to the information of the spatial location of the events. Thus, we will consider our sample as a unique spatial region for the analysis. Every event occurring at time tMS with energy EMS is considered to be a mainshock. We study the activity rate rAS (number of events per time interval) as a function of the time after the MS, ∆t=t−tMS, until an event with energy E > EMS is found. This indicates the end of the aftershock sequence. Averages of the AS rates ⟨rAS(∆t)⟩are performed by considering all the sequences corresponding to MS within the windows Ei−1< EMS < Ei. One then divides the time axis in (logarithmic) bins and counts the number of AS in every bin, from all sequences, and normalizes it by the number of sequences reaching that bin. The results of this analysis are shown in Fig. 4. The legend indicates the different windows (Ei−1, Ei) for EMS. The time ∆t=t−tMS has been transformed into a temperature increase by using ∆T=R∆twhere Ris the heating rate (RAE or Rcal). The minimum ∆Tvalues corresponding to the aftershock sequences obtained by AE and calorimetry are separated by one order of magnitude. This is a consequence of the difference in statistics between both measurement techniques. Both AE and calorimetric data show a slight but clear decay in the production of aftershocks with an exponent Avalanche correlations in the martensitic transition 9 10-1 100 101 102 10-3 10-2 10-1 100101102103 <rAS> (DT) (mK-1) DT (mK) p=0.15 AE 0.64<E(aJ)<1.28 1.28<E(aJ)<2.56 2.56<E(aJ)<5.12 5.12<E(aJ)<10.24 10.24<E(aJ)<20.48 20.48<E(aJ)<40.96 40.96<E(aJ)<81.92 cal 0.08<E(mJ)<0.16 0.16<E(mJ)<0.32 0.32<E(mJ)<0.64 0.64<E(mJ)<1.28 1.28<E(mJ)<2.56 2.56<E(mJ)<5.12 Figure 4. Comparison between the production of aftershocks in both Acoustic Emission and Calorimetry data as a function of the temperature variation after a mainshock ∆T. The straight line shows the power law behavior corresponding to an exponent p= 0.15. Note that if the avalanches were fully uncorrelated these plots will be flat. p∼0.15 and a negligible dependence on EMS. This may suggest a very low productivity exponent α. The immediate conclusion deduced from this result is the existence of weak correlations between avalanches. Thus, a local Poisson process is expected to be unsuitable to depict the avalanche process during the MT. 4.2. Bi-test In order to get a more robust proof of the previous results we endeavor to show that the studied avalanche processes are non-Poissonian, even when one considers small time windows in order to correct for the non-stationary behavior. We follow Bi, B¨orner and Chu [18] who introduced a method to refute the null-hypothesis of local Poissonity in a point process. Consider the sequence of events at {tk}depicting a point process. The Bitest is based on the joint evaluation of the waiting time to the closest event δtk= min{tk−tk−1, tk+1 −tk}and the consecutive waiting time δτkin the same temporal direction, i.e.: δτk=tk−1−tk−2if δtk=tk−tk−1or δτk=tk+2 −tk+1 if δtk=tk+1 −tk. From the data pairs {δtk,δτk}we can build the statistic variable Hk=δtk δtk+1 2δτk ,(2) that takes values between 0 and 1.